Question

Исследуйте функцию y=|ctgx|+cosx на периодичность укажите основной период если он существует. Помогите пожалуйста

Answer 1

Это периодическая функция. Период 2 пи.
1. |ктгХ| - периодическая
2. косХ - периодическая
3. их периоды соизмеримы
из (1-3) => их сумма тоже периодическая
наверно...
для частного случая легко доказывается

Answer 2

Function Analysis: y = |ctgx| + cosx

To analyze the function y = |ctgx| + cosx, let's break it down step by step.

The function consists of two parts: |ctgx| and cosx. Let's analyze each part separately.

1. Analysis of |ctgx|: The function |ctgx| represents the absolute value of the cotangent of x. The cotangent function is defined as the ratio of the adjacent side to the opposite side of a right triangle. However, the cotangent function is undefined at x = kπ, where k is an integer, because the opposite side becomes zero. To avoid this undefined behavior, we take the absolute value of the cotangent function, which ensures that the function is always positive.

2. Analysis of cosx: The function cosx represents the cosine of x. The cosine function is periodic with a period of 2π. This means that the function repeats itself every 2π units.

Now, let's analyze the periodicity of the function y = |ctgx| + cosx.

Periodicity of y = |ctgx| + cosx

To determine the periodicity of the function y = |ctgx| + cosx, we need to consider the periodicity of each component separately.

1. The function |ctgx| is periodic with a period of π. This means that the function repeats itself every π units. The absolute value operation does not affect the periodicity of the function.

2. The function cosx is periodic with a period of 2π. This means that the function repeats itself every 2π units.

To find the overall periodicity of the function y = |ctgx| + cosx, we need to find the least common multiple (LCM) of the periods of the individual components.

The LCM of π and 2π is 2π. Therefore, the function y = |ctgx| + cosx is periodic with a period of 2π.

Conclusion

In conclusion, the function y = |ctgx| + cosx is periodic with a period of 2π. This means that the function repeats itself every 2π units.

I hope this helps! Let me know if you have any further questions.